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Theorem eqvrelsymb 35721
Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)
Hypothesis
Ref Expression
eqvrelsymb.1 (𝜑 → EqvRel 𝑅)
Assertion
Ref Expression
eqvrelsymb (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem eqvrelsymb
StepHypRef Expression
1 eqvrelsymb.1 . . . 4 (𝜑 → EqvRel 𝑅)
21adantr 481 . . 3 ((𝜑𝐴𝑅𝐵) → EqvRel 𝑅)
3 simpr 485 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3eqvrelsym 35720 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 481 . . 3 ((𝜑𝐵𝑅𝐴) → EqvRel 𝑅)
6 simpr 485 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6eqvrelsym 35720 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 797 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   class class class wbr 5057   EqvRel weqvrel 35351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-refrel 35632  df-symrel 35660  df-trrel 35690  df-eqvrel 35700
This theorem is referenced by:  eqvrelth  35726
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